Upper bounds for the number of zeroes for some Abelian integrals
Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2012
Abstract:	Consider the vector field x' = -yG(x, y), y' = xG(x, y), where the set of critical points {G(x, y) = 0} is formed by K straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree n and study which is the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of K and n. Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and in a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for K ≤ 4 we recover or improve some results obtained in several previous works.
Grants:	Ministerio de Ciencia e Innovación MTM2008-03437 Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-410 Ministerio de Ciencia e Innovación MTM2009-06973 Agència de Gestió d'Ajuts Universitaris i de Recerca 2009/SGR-859
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Document:	Article
Subject:	Abelian integrals ; Weak 16th Hilbert's Problem ; Limit cycles ; Chebyshev system ; Number of zeroes of real functions
Published in:	Nonlinear Analysis : Theory, Methods and Applications, Vol. 75 (2012), p. 5169-5179, ISSN 0362-546X

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